Mathematics – Classical Analysis and ODEs
Scientific paper
2010-09-07
Mathematics
Classical Analysis and ODEs
To appear in the J. Geom. An
Scientific paper
We obtain sharp bounds for the modulus of continuity of the uncentered maximal function in terms of the modulus of continuity of the given function, via integral formulas. Some of the results deduced from these formulas are the following: The best constants for Lipschitz and H\"older functions on proper subintervals of $\mathbb{R}$ are $\operatorname{Lip}_\alpha ( Mf) \le (1 + \alpha)^{-1}\operatorname{Lip}_\alpha( f)$, $\alpha\in (0,1]$. On $\mathbb{R}$, the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le (\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the asymptotic behavior, as $d\to\infty$, of the norm of the maximal operator associated to cross-polytopes, euclidean balls and cubes, that is, $\ell_p$ balls for $p = 1, 2, \infty$. We do this for arbitrary moduli of continuity. In the specific case of Lipschitz and H\"older functions, the operator norm of the maximal operator is uniformly bounded by $2^{-\alpha/q}$, where $q$ is the conjugate exponent of $p=1,2$, and as $d\to\infty$ the norms approach this bound. When $p=\infty$, best constants are the same as when $p = 1$.
Aldaz J. M.
Colzani Leonardo
Lázaro Pérez J.
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